Question: Tom started an entertainment company. The net value of the company (in thousands of dollars) $t$ months after its creation is modeled by $v(t)=4t^2-24t-28$ Tom wants to know when his company will be at its lowest net value. 1) Rewrite the function in a different form (factored or vertex) where the answer appears as a number in the equation. $v(t)=$ 2) How many months after its creation does the company reach its lowest net value?
Solution: Choosing a form The company's lowest net value relates to the minimum of the function. Which form reveals this feature? Here's a summary of what each form reveals along with examples. Note that these are all equivalent forms of the same function, but not the function modeling the net value of the company. Form Example Feature revealed Standard $f(x)=2x^2-12x+{10}$ $y$ -intercept is ${10}$ Factored $f(x)=2(x-C{1})(x-C{5})$ Zeros are $x=C1$ and $x=C5$ Vertex $f(x)=2(x-{3})^2{-8}$ Vertex is $(3,{-8})$ Rewrite in vertex form The vertex of the function tells us the value of $t$ where the function reaches its minimum value, so let's rewrite $v(t)$ in vertex form by completing the square. The number that will help us complete the square is $\left(\dfrac{{-6}}{2}\right)^2={9}$ : $\begin{aligned} v(t)&=4t^2-24t-28 \\\\ &=4(t^2-6t)-28&&\text{Factor } 4 \text{ from first two terms}. \\\\ &=4(t^2{-6}t+{9})-28{-36}&&\text{Complete the square}. \\\\ &=4(t-3)^2-64&&\text{Factor and simplify}. \end{aligned}$ [How do we know what to add to complete the square?] When does the company reach its lowest net value? The vertex form of the function reveals its vertex, and we know this point is a minimum for $v(t)$ since the leading coefficient $4$ is positive. In general, for any quadratic function written in vertex form $f(x)=a(x-{h})^2+{k}$, we can conclude that the vertex is the point $({h},{k})$. So for $v(t)=4(t-{3})^2{-64}$, the vertex is $({3},{-64})$, and we know the company reaches its lowest net value at $t={3}$ months. Answers 1) The vertex form of the function reveals when the company reaches its lowest value: $v(t)=4(t-3)^2-64$ 2) The company reaches its lowest net value $3$ months after creation.